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Exemplar Unit Plan
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Brief Description Uses
Idea Generation
Allow this unit plan and others in the bank to serve as idea generators for yourself and your unique setting.
Pick and choose components of the unit plan which serve you well and use them as you see fit.
Adopt the format of the plan while replacing all of the components with more appropriate material for your specific setting.
Implementation
Take and use the unit plan in your specific setting. Confirm with administrators and instructional support staff that the plan fulfills the needs your classroom has.
Implement unit plan in your classroom as it is planned in the document. Use accompanying materials prescribed and objectives listed.
GENERAL UNIT INFORMATION
Grade/Subject: 10th Grade/ Advanced Algebra 2
Unit Name: Polynomials and Polynomial Functions
Dates of Unit
Implementation: October 22– November 9
UNIT STANDARD(S)
Common Core State Standards
Standard(s)
HSA-SSE.A.1a - Interpret parts of an expression, such as terms, factors, and coefficients.
HSA-SSE.B.3c - Use the properties of exponents to transform expressions for exponential
functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal
the approximate equivalent monthly interest rate if the annual rate is 15%.
HSA-APR.A.1 - Understand that polynomials form a system analogous to the integers, namely,
they are closed under the operations of addition, subtraction, and multiplication; add, subtract,
and multiply polynomials.
HSA-SSE.A.2 - Use the structure of an expression to identify ways to rewrite it.For example, see
x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 –
y2)(x2 + y2).
HSA-APR.B.3 - Identify zeros of polynomials when suitable factorizations are available, and
use the zeros to construct a rough graph of the function defined by the polynomial.
HSA-APR.B.2 - Know and apply the Remainder Theorem: For a polynomial p(x) and a number
a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
Remedial
Standards
HSA-SSE.A.1b - Interpret complicated expressions by viewing one or more of their parts as a
single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on
P.
HSA-SSE.B.3a - Factor a quadratic expression to reveal the zeros of the function it defines.
Note – there are many more remedial standards, however, these would be the most recently
learned and relevant standards
Enrichment
Standards
HAS-BF.B.3 & 4 – building new functions from existing functions could be an enrichment,
however, these standards are also explicitly taught in the next unit
Students can also be given extension problems where they must, without guidance, try to build a
possible function given the graph of a polynomial.
Curricular
Resources
Geogebra.org – free graphing utility that students and/or teacher can use to investigate the
connection between the polynomial expressions and their graphs.
Demonstrations.wolfram.com – site contains visuals that can show the transformation of
polynomials and their graphs. Search for “polynomial”.
‘Big goal’, Assessment and Investment
BIG GOAL
Qualitative:
Students will develop and be able to articulate their deep understanding of the connection between an
algebraically expressed function and its graph including an understanding of roots of an equation and their
representation on a graph. Students will develop the skills necessary to find the zeros of a polynomial
equation and use their knowledge to sketch a graph of a polynomial.
Quantitative:
At least 80% of students will score at least 80% on the unit quizzes and unit test.
All students will earn at least a 90% on the Exponent Qualifier with in the first two attempts.
FORMATIVE & SUMMATIVE ASSESSMENT
How will I measure PROGRESS towards the Unit Goal?
Students will be given daily warm-ups that will be used as a quick informal check for understanding.
Many of these will address the qualitative goals of the unit.
Students’ homework assignments will also be checked during the warm up. The teacher will choose one
problems she anticipates to have been challenging and will check that problem as she checks homework.
Students will also be given regular exit tickets one 1 – 3 quick problems as a check for understanding.
These tickets also will often include a computational question as well as a question requiring concept
comprehension and a written response. Many of the concept questions will address the qualitative goal
of the unit.
There will be one unit quiz that counts toward students’ grades (see Calendar – Quiz 6A)
How will I measure ACHIEVEMENT of the Unit Goal?
Students must pass a “qualifying” exam for their exponent rules. Students will have two opportunities
to pass this exam with at least a 90% or higher. Students will be given one chance in class and will be
required to attend tutoring and complete additional practice if they do not pass with a 90%. Only after
tutoring and additional practice will they be allowed to re-take the exam with a 10% grade deduction for
each additional try.
Note: Our vertical alignment team has discovered that not knowing exponent rules well enough at this
level will cause enormous challenges later on in this course as well as every subsequent math course. I
have implemented this strict qualifying exam for two years and all students have passed by the second
time. I also send home a letter to parents explaining the severe expectations for this exam. The letter is
included after the exams. Each time a student takes the qualifier he or she receives a different form of
the exam.
Students will also complete a unit exam that will cover all the objectives, except for the exponent rules.
INVESTMENT PLAN SUMMARY (*cut & paste acceptable)
Engagement Students will be given an assignment to investigate their own thoughts about what the graph of a
function truly represents. Although we have discussed this at the beginning of the school year with linear,
absolute value, and linear piecewise functions, they have not investigated this question in detail with curves.
Students will have to answer the following questions over the course of the three state test days and
then we will discuss their ideas when they return at the end of the first week of the unit (see Calendar for more
clarification).
What about an equation causes its graph to be curved? Why do you think this happens?
Why might it be important to know where a function will cross the x-axis? Consider a function that
represents your money in a stock where the x-axis represents time and the y-axis represents the amount
of money you have. What does the x-axis actually represent in this situation?
How do you find the x-intercept of a linear function? How would you find the x-intercept of a non-
linear function? How are these related?
What does the graph of an equation represent?
Complete the table below for the given function and then plot each point on the graph below. What do
you think the graph looks like beyond the points your plotted? Sketch, using a pencil, your prediction.
Students will be asked to share their findings with a partner when they return at the end of the week and
will be given the opportunity to share their thoughts with the class. The teacher will not directly answer any of
their questions at this time, but will announce that they will have concrete answers throughout the unit.
Class Tracking The class averages and mean values of the assessments will be posted when the assessments are returned to
students.
Individual Tracking
Students keep a sheet where they track their assessment scores so that they can see their progress.
Student Reflections On the back or at the end of some of the exit tickets, students will describe how they feel about the unit and what
they are learning. Depending on the comment, the teacher may write a response to be handed back the next day.
UNIT SUMMATIVE ASSESSMENT
ALIGNMENT GUIDE
The assessments are included at the end of the unit plan. T represents the unit test and Q represents the
Exponent Qualifier.
Standard # Standard
Aligned to
item #’s
Points
Possible
HSA-SSE.B.3c
Use the properties of exponents to transform expressions for
exponential functions. For example the expression 1.15t can be
rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate
equivalent monthly interest rate if the annual rate is 15%.
Q1-Q30 30
HSA-APR.A.1
Understand that polynomials form a system analogous to the
integers, namely, they are closed under the operations of
addition, subtraction, and multiplication; add, subtract, and
multiply polynomials.
T5, T6, T7 3
HSA-SSE.A.2
Use the structure of an expression to identify ways to rewrite it.
For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a
difference of squares that can be factored as (x2 – y2)(x2 + y2).
T8 - T11 8
HSA-APR.B.3
Identify zeros of polynomials when suitable factorizations are
available, and use the zeros to construct a rough graph of the
function defined by the polynomial.
T3, T4, T19-T20 4
HSA-APR.B.2
Know and apply the Remainder Theorem: For a polynomial
p(x) and a number a, the remainder on division by x – a is p(a),
so p(a) = 0 if and only if (x – a) is a factor of p(x).
T1, T12-T18 15
HSA-SSE.A.1a Interpret parts of an expression, such as terms, factors, and
coefficients. T2 1
Total points possible: 61
***Assessment outcome data needed for final presentation!
Lesson Objectives
(*A.K.A. : ’Breaking it down, day-by-day’)
Standard
List the main standards
to be taught here.
Note: The following
standard is too complex
to be taught in its
entirety on the same
each day. Therefore, it
will be repeated BUT,
broken down into daily
objectives.
Daily Lesson Objective(s)
SWBAT
Timing
Most lessons will
be taught for 55
min. /day
A total of 10 days
will be used to
teach the daily
lesson objectives
in order to meet
the unit goal.
Summarized (yet specific )Lesson Plan Notes
This area is to include ideas you have for teaching the lesson. Include strategies,
materials and potentially, procedures.
HSA-SSE.B.3c
simplify expressions using the first four
properties of exponents (see Exponent
Qualifier)
Students will already be familiar with many of these rules, so these can be taught
through inquiry. Students can be lead through investigating the basic meaning of
an exponent as a “rapid” form of multiplication and then they can write their own
rules. Students can work in groups to do this and can share their rules on
designated spaces around the room.
Simplify expressions using the last four
properties of exponents (see Exponent
Qualifier)
Students can use what they learned the previous day to expand their rules. The
first three will be straight forward and the students can investigate these by
themselves for a few moments and then they can review them whole group.
The last rule is often difficult for students to think about so this will be more
direct instruction but the teacher should have students guess at what fractional
exponents mean.
HSA-APR.A.1
HSA-SSE.A.1a
Label polynomials based on their degree
and number of terms.
Use arithmetic so simplify polynomial
expressions
Students will probably have questions on the previous lesson after having done
their homework, so almost half of the class time should be devoted to that. The
objectives for this day are very quick and students should be quite familiar with
simplifying polynomial expressions. Students can play a game to help memorize
and recognize the names of the polynomials.
HSA-SSE.A.2
Factor polynomials including the sum and
difference of two cubes and expressions
that are “like quadratics”
Students have already factored quadratics and by grouping in the previous unit,
so most of this standard has already been taught. Students can be presented with
several different types (see below) of polynomials and will be asked to put them
into different groups and to figure out which ones they know how to factor and
which ones they do not know how to factor. As a class, students can make
suggestions for how we could factor the types they are unfamiliar with. (As
foreshadowing, the warm-up for the day should be multiplying a binomial and
trinomial that simplify to the sum or difference of two cubes.)
HSA-APR.B.3
Use synthetic division to divide, factor,
and find the roots of polynomials given a
root or factor.
Ensure that students know the difference between a root and a factor of a
polynomial. Review their initial engagement activity and provide some light
practice in defining roots and factors. Begin the lesson by reviewing long
division and having students write out steps as if they were going to describe
these to someone else. Apply their steps to long division of a polynomial and
then blow their minds by circling the coefficients, then doing the same problem
with synthetic division. (Color coding would probably be very helpful here;
perhaps highlighting the coefficients as you go when doing the synthetic
division.)
Find zeros using the fundamental theorem
of algebra and write polynomial functions
given roots
In between the last objective and this one, students should learn the Rational
Root Theorem (HAS-APR.B.2)
Students should be directed back to their engagement activity and should review
and/or revise their thoughts. This lesson should include a lot of conceptual
checks for understanding and students should be asked to explain their reasoning
and explain the connection between zeros and the equation of a polynomial
function.
Sketch and analyze the graphs of
polynomials given different pieces of
information
55-75 min.
between two days
Students should discuss the accuracy of their sketches and what information they
would need to make their sketches more accurate. As an extension for all,
students should be asked at the end of the lesson to write the equation of a
polynomial given its graph. All students should attempt this.
Write a possible equation of a polynomial
given its graph This is a very short lesson and may not consist of a whole class period.
HAS-APR.B.2 Find rational zeros by using the rational
root theorem.
Students can be given a set of polynomials and their roots and asked to see if they
can find a pattern between the roots and any set of the coefficients of the
polynomials. The teacher should ask if the students can find a short cut or “way
to cheat” to find the roots simply by looking at the equation.
Total days/minutes taught during the entire unit of instruction. 495 - 550 min. days 9 - 10
Real-Time Calendar
Date YWBAT:
Mon. 10/22 (6.1A) simplify expressions using properties of exponents (rules 1 – 4)
Tue. 10/23 State Testing in the morning, then classes:
Tue.: Per. 1 & 2, Wed.: Per. 3 & 4, Thur.: Per. 5 & 6
(6.1B) simplify expressions using properties of exponents (rules 5 – 8) Wed. 10/24
Thur. 10/25
Fri. 10/26 (6.3) name and simplify polynomial expressions using arithmetic
Mon. 10/29 (6.4) factor polynomial expressions including the sum and difference of two cubes
Tue. 10/30 (6.5) use synthetic division to divide, factor, and find the roots of polynomials, given a root or factor.
Wed. 10/31 (6.6) find rational zeros by using the rational root theorem
Thur. 11/1 (6.7) find zeros using the fundamental theorem of algebra and write polynomial functions given roots
Fri. 11/2 Quiz 6A (6.1 – 6.6)
(6.8) sketch the graphs of polynomials (if time permits)
Mon. 11/5 (6.8) sketch and analyze the graphs of polynomials
Tue. 11/6 Chapter 6 Review (focus on 6.1 – 6.4)
Wed. 11/7
Early Release (38 min.)
Exponent Qualifier: You must earn a 90% or higher. If you do not, you have one chance to re-take it and
again must earn at least a 90% but will get a 10% reduction.
Thur. 11/8 Chapter 6 Review (focus on 6.5 – 6.8)
Fri. 11/9 CHAPTER 6 TEST
Chapter 6 Test
Lawson Risoldi, Adv. Alg. 2 (2011), Pg. 1
Chapter 6 Test Name ____________________________________
QUADRATIC FUNCTIONS Date __________ Period ______
DIRECTIONS: Write neatly, use pencil, blue, or black ink. Show your work and simplify all answers.
PART A: Find the best answer. Write your answer in the space provided. (1 point each)
_____ 1. Evaluate 3 2( ) 2 4 5 8f x x x x at 2x .
(A) 2 (B) 18 (C) 8 (D) 34 (E) 50
_____ 2. Which of the following expressions is a quadratic binomial?
(A) 4 2x x (B)
3x x (C) 24x (D)
2 5x x (E) None of these
_____ 3. Which of the following is a possible root of3 2( ) 4 5 9f x x x x ?
(A) 1
3 (B)
1
9 (C) 18 (D) 9 (E) None of these
_____ 4. Which function below could have the graph shown in Figure A?
(A) 2x (B) 31
2x (C)
4x
(D) 32x (E) None of these
_____ 5. Simplify 3 24 4 3 9x x x x .
(A) 3 24 2 13x x x (B)
3 24 4 5x x x (C) 3 24 2 13x x x
(D) 4 3 28 13 3 36x x x x (E) None of these
_____ 6. Simplify 24 4 9x x x .
(A) 9
4x
x
(B)
2 8 7 36x x x (C) 3 28 7 36x x x
(D) 4 316 7 36x x x (E) None of these
_____ 7. Simplify 5 6 2x x x .
(A) 3 29 8 60x x x (B)
3 29 8 28x x x (C) 3 60x
(D) 3 60
9 85
x xx
(E) None of these
Figure A
Chapter 6 Test
Lawson Risoldi, Adv. Alg. 2 (2011), Pg. 2
PART B: Factor the following polynomials. Write your answers in the answer box. (2 points each)
8. 4 16x 9.
3 25 9 45x x x
10. 327 1x 11.
3 24 6 10x x x
PART C: Simplify the following expressions. Write your answers in the answer boxes. (2 points each)
12.
3 23 4 5
2
x x
x
13. 4 3 22 5 1 1x x x x x
PART D: Factor the following polynomials with the given information. Place your answers in the boxes below.
(2 points each)
14. 3 23 13 15x x x , given ( 5)x is a factor. 15.
3 25 2 24x x x , given ( 2)x is a factor.
ANSWER BOX
8.
9.
10.
11.
Chapter 6 Test
Lawson Risoldi, Adv. Alg. 2 (2011), Pg. 3
PART E: Given a zero of the polynomial, find the remaining zeros. Write your answers in the box below.
(2 points each)
16. 3 23 4 12x x x , given 3 is a zero.
PART F: Find all the zeros of the function. Write your answer in the boxes below. (2 points each)
17. 3 2( ) 9 10 17 2f x x x x 18.
4 3 2( ) 7 13 3 18f x x x x x
Chapter 6 Test
Lawson Risoldi, Adv. Alg. 2 (2011), Pg. 4
PART G: Follow the directions for each. (1 point each)
19. Sketch the graph of ( ) 2 2 3f x x x x 20. Write a possible function in factored form of the
function graphed below.
CHALLENGE: You must complete all previous problems in order to receive any extra credit. Box your answers on
this page.
21. Write the equation of the function shown in the graph below. (Hint: Use a piecewise function) (1%)
Exponent Qualifier
Lawson Risoldi, Adv. Alg. 2 (2011) , Pg. 1
Exponent Qualifier
Name ____________________________________
FORM A Date __________ Period ______
PART A: Complete the following expressions. Write neatly. (2 points each – however, you must get all of these
correct in order to “pass”.)
PART B: Simplify the following. Write your answers with positive exponents and in radical notation where
applicable. Write your answers neatly in the answer box. You do not have to rationalize the denominator.
(1 point each)
9. 3 5x x 10.
42 3x y 11.
27 12.
34x
13.
5
3 2
4
16
xy
x y 14.
23
10
5x
y
15. 0 2 3 53 x y x
16. 3/25 17.
61/3 5/6x y 18.
22
5 4
7xy
x y
1. x ya a 3. x
ab 5. xa 7. y
xa
2. x
y
a
a 4.
xa
b
06. a /8. x ya
Answer Box
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
Exponent Qualifier
Lawson Risoldi, Adv. Alg. 2 (2011) , Pg. 2
19.
20
2
x y
xy
20. 3/2x
21. 3
2x
22. 3/2 5/2x x 23.
27
3
24. 0 2
2 7 345 3x y z x y
25. 3
1/5 9/5x x 26.
1/3
1/6
y
y 27.
3/22
28.
2 1
3 0
2
4
x y
x y
29.
12 2 23x y z
30.
31
x
Answer Box
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
Dear Parent,
I am writing to inform you of a very important test your child will take in this unit of Advanced
Algebra 2. This unit covers a skill that is critical to students’ success later on in this course and nearly
every other math course they will take. Because of this, I am requiring that students pass a test on this
topic with no lower than a 90% because that is the level of proficiency they will need to be successful
later on.
Students will have two chances to take this exam and will receive a 10% score deduction for
taking it more than once. I employ this 10% penalty to encourage students to study well and get the
help they need to do well the first time.
If students do not pass with 90% the first time, they will receive no score in the gradebook and
will be required to attend tutoring and complete additional practice before re-taking the exam. I am
available before and after school for tutoring, but students should check the tutoring schedule to before
attending.
I have required this exam for the last two years and every student has passed within two tries. I
am confident that all students will pass again this year. I have also seen significant improvement in
students’ work in later chapters and courses after implementing this qualifying exam. This year, all of
the Advanced Algebra 2 teachers are giving this exam.
Please reach out to me with any questions or concerns you might have.
Sincerely,
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